US7451415B2 - Method for predicting inductance and self-resonant frequency of a spiral inductor - Google Patents
Method for predicting inductance and self-resonant frequency of a spiral inductor Download PDFInfo
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- US7451415B2 US7451415B2 US11/435,710 US43571006A US7451415B2 US 7451415 B2 US7451415 B2 US 7451415B2 US 43571006 A US43571006 A US 43571006A US 7451415 B2 US7451415 B2 US 7451415B2
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- inductor
- inductance
- self
- resonant frequency
- spiral inductor
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/30—Circuit design
- G06F30/36—Circuit design at the analogue level
- G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/30—Circuit design
- G06F30/36—Circuit design at the analogue level
Definitions
- This invention relates to a method for predicting self-resonant frequency and inductance of two-port on-chip freely suspended spiral inductor.
- This prior invention provides a method for optimally designing an inductor of inductor circuit.
- an equivalent circuit of the inductor or inductor circuit will be presented.
- a posynomial expression will be obtained for each lumped parameter.
- quality factor or self-resonant frequency can be shown in posynomial form.
- Geometric program in posynomial form can be written to constrain on inductor performance specifications. This method can be utilized to optimally design inductors circuits.
- This prior invention provides a method and integrated design system for predictive layout generation for inductors with reduced circuit design cycle.
- the invention receives a number of parameters for an inductor, such as Number of Turns, Spacing, Width, Xsize, and Ysize, to determine parasitic resistor values and parasitic capacitor values which are used in simulating the circuit model comprising the inductor.
- an inductor layout is then generated that results in those parasitic values.
- This prior invention provides a generic method for simulating electromagnetic field problems, and is designed for solving solution of partial differential equation by numerical method under condition of numerical stability.
- this prior invention introduces a dummy field for describing the underlying physical phenomena of non-uniqueness of the electric field and magnetic potentials. A special caution should be taken in the translation of the continuous field equation onto the discrete, comprising of nodes and links.
- the prior invention provides a method for numerical analysis by directly solving the field equation modified by addition of a dummy field.
- the dummy field is preferably a scalar field.
- a least one parameter relating to a physical property of physical system will be outputting, such as field strength, a resistivity, an inductance, an energy value, etc. It also includes a step refining a mesh used in the numerical analysis in accordance with an embodiment of the prior invention.
- This prior invention may provide an apparatus and include a data structure and computer program for use in numerical analysis of a simulation of a physical system.
- the field equations can be solving easily by addition of the dummy field and a representation of an n-dimensional mesh in a predetermined domain of the physical system.
- this prior invention is absolutely an excellent method for simulating the frequency dependence of characteristic of electromagnetic field.
- the time-consumption is an irremovable issue for a simulation tools and methods.
- the model of the present invention provides a method for saving the wasted hours.
- the present invention provides a method could not only solving the problem of the time-consumption but also evaluating the self-resonant frequency and inductance of inductor without complicated analysis.
- This paper is actually a bible for designing microelectronic inductor. Unlike others, this paper points out the effect of negative mutual in the scale in the microelectronic world. Inductance of rectangular inductors can be evaluated by the geometric factors, such as track width, space between tracks, and number of turns. The frequency dependence of inductance and self-resonant frequency, however, can not be obtained by this paper and the complicated geometric analysis will confuse the engineers and designer very mush.
- the model of the present invention can handle this situation.
- the self-resonant frequency can be easily calculated by physical based close form.
- the frequency dependence of the inductance can also easily calculate without complicated geometric analysis.
- the model of the present invention should simplify the work of complicated geometric analysis that Greenhouse even done and have more physical sense.
- the high frequency simulation system can solve any shape and scale devices in high frequency domain by evaluating the electromagnetic matrix. The parameters then can be output to calculate the characteristic values.
- time-consumption is a great issue here. More geometry complicates, more hours needs.
- the model of the present invention can save those waste CPU time.
- the present invention address a method to predict inductance and self-resonant frequency of two-port on-chip freely suspended spiral inductors.
- This invention can easily evaluate the inductance and self-resonant frequency of polygon spiral inductors without complicated geometry analysis, reduce the time-consuming rather than the computer simulations, and save money and manpower when industries investigate RF inductor and inductor circuits, such as cell phones, communication systems, etc. when testing.
- the present invention can also solve the problems of complicated circuit analysis.
- One of the purposes of the present invention is to determine the self-resonant frequency. Furthermore, the model of the present invention can explain the reasons that difference of the self-resonant frequency of different material with physical sense in which designers can optimally analyze products. Then the process of the posynomial expression might be not easy to understand and follow actually. Unlike the inconvenience of prior methods, the model of the present invention can show simple steps from evaluating the self-resonant frequency to determine the inductance of inductors. Fortunately, the model of the present invention also can determine the self-resonant frequency with different shape, such as circular, octagonal, and rectangular inductors.
- the present invention not only considers the geometric parameters of the inductor but also the material characteristics.
- the present invention can avoid the complicated geometric analysis and presents the self-resonant frequency and the inductance associated with frequency with serious mathematical and physical sense.
- the present invention can evaluate the characteristic values of a spiral inductor without confusing circuit design, and also provide engineers and designers for optimizing the characteristics of the inductors.
- the present invention provides a novel and unsophisticated method to accurately evaluate the inductance and self-resonant frequency of RF on-chip spiral inductors. Differencing from the methods of the prior arts to calculate the inductance of a spiral inductor, the present invention evaluates the greater part of inductance of an inductor while altering the material of the inductor by means of modifying parameters of formulations in this invention. Thus, this invention provides a method to evaluate the self-resonant frequency of a spiral inductor. The determination of the self-resonant frequency helps the circuit designers and the microwave engineers to choose the appropriate inductance bandwidths.
- the Kramers-Kronig relations compose one of the most elegant and general theorems in physics because their validity only depends on the principle of causality: the response cannot come before the stimulation. Thus, the relations will be so powerful to analyze the conjugate mathematical and physical phenomenon. Based on the Riemann-Lebesgue lemma, the characteristic of a conducting medium, and the electromagnetic field theory, the Kramers-Kronig relations have the form for real and imaginary parts:
- ⁇ , ⁇ B , n e and m e are Planck's constant, Bohr magneton, the free electron density and mass in material, respectively.
- the first and second terms in (3) are the paramagnetism and diamagnetism factors of the inductor material.
- the symbol [n ⁇ 1] represents the Gaussian symbol where n is the number of turns.
- the parameters, l max and s, represent the maximum edge and line spacing of the inductor, respectively.
- E L 3 / 5 ⁇ ⁇ 2 m e ⁇ ( 3 ⁇ ⁇ ⁇ 2 ⁇ n e ) 1 / 3 ⁇ ⁇ l total ( 4 )
- l total represents the total length of the inductor.
- E ⁇ ⁇ ( r ) 1 4 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 0 ⁇ q ⁇ [ ⁇ + 8 ⁇ ⁇ sin ⁇ ( ⁇ 2 / 4 ⁇ ⁇ ⁇ ) ] 2 8 ⁇ ⁇ ⁇ ⁇ ⁇ h ⁇ ⁇ ⁇ ⁇ [ csc ⁇ ( ⁇ / 2 ) - 1 ] ⁇ ( ⁇ + ⁇ ) ⁇ r ⁇ ( 5 )
- q is elementary charge ( ⁇ 1.6 ⁇ 10 ⁇ 19 C)
- ⁇ and h are width and height of the spiral inductor, respectively.
- ⁇ is the corner angle, and the field is centered at the outer apex of each corner.
- the SRF would be changed and can be calculated in (6) by changing the integral range (r m , ⁇ ) to replace ( ⁇ , ⁇ ), where the factor r m presents a reference point for an inductor circuit. This factor indicates the loss or shift term for applying energy.
- the inductance can be derived with the associated magnetic energy of EM field in the inductor:
- model of the present invention could predict the self-resonant frequency and inductance of a on-chip freely spiral inductor and the designer could easily satisfy their requirements by means of altering the geometry and material of their inductors.
- the analytical method based on Kramers-Kronig relations and EM field theory could provide us mathematically convenience for the inductor design in physical sense.
- FIG. 1 A block flow chart illustrating the method of the present invention.
- FIG. 2 The schematic diagram of the two-port spiral polygon inductor realizations: (a) rectangular, (b) octagonal, (c) circular. lmax, S, and ⁇ are the maximum edge, line spacing, and line width of the polygon inductor, respectively.
- the frequency range is from 1 GHz to 20 GHz
- the frequency range is from 1 GHz to 9 GHz.
Abstract
Description
where σ0 is the dc conductivity of metal, ω is the frequency of electromagnetic (EM) field in the system, and ωr is the self-resonant frequency. Then, the parameter α has the from:
where η, μB, ne and me are Planck's constant, Bohr magneton, the free electron density and mass in material, respectively. For a spiral inductor with the geometry as shown in
B. Determination of the Self-Resonant Frequency:
where ltotal represents the total length of the inductor. According to electromagnetic field theory, there will be electric fields built up in the neighborhood of corners while an external electric field is applied on a conducting material. Thus, for a polygon spiral inductor, the quasi-electrostatic electric field built up in each corner has the form as the following that is calculated by variation principle:
where q is elementary charge (˜1.6×10−19 C),
where r, N, V, and σeff are the electronic incident path, the number of corners, the volume of polygon spiral inductor, and the effective cross section of the inductor, respectively. Here, the effective cross sections are equal to 0.101, 0.281, and 0.375 times the cross section, A, of rectangular, octagonal, and circular inductors (are shown in
means the trajectory function with the perturbation terms. The first term means the ideal trajectory function, and the second terms means perturbation from the near field such as the ground pad, etc. Since we assume the perturbations from the field is far from the infinity, the integral arguments are as the presentation. The recent investigations reveal that the substrate coupling effects could be neglected as long as the air gap is larger than 60 μm. For a micromachined inductor in the RFIC design, the reference ground point would be far away from the inductor. Thus, infinity assumption is reasonable and practical in the model. Nevertheless, if a reference ground plane is designed to close to the inductor, the SRF would be changed and can be calculated in (6) by changing the integral range (rm, ∞) to replace (−∞, ∞), where the factor rm presents a reference point for an inductor circuit. This factor indicates the loss or shift term for applying energy.
ωr=(E L +N·E C)/η (8)
The electron energy is equal to the kinetic energy plus the total energy lost in the corner field scattering.
C. Determination of the Inductance
where kB and T are the Boltzmann's costant and absolute temperature, respectively. The free electron density and conductivity in above formula indicated that the inductance shall depend on the characteristic of material seriously.
D. Consider a Real Case:
TABLE 1 |
SELF-RESONANT FREQUENCY WITH |
DIFFERENT TYPE OF INDUCTOR |
Self-resonant frequency | ||
Comparisons | based on the model of the | Self-resonant frequency |
(n = 3.5) | present invention (GHz) | based on HFSS (GHz) |
Rectangular | 23.9 | 22.9 |
inductor | ||
Octagonal | 24.9 | 23.6 |
inductor | ||
Circular | 25.8 | 24.6 |
inductor | ||
TABLE 2 |
COMPARISON OF INDUCTANCE |
OF RECTANGULAR INDUCTOR |
Self-resonant | Inductance | Inductance | Inductance | |
Comparisons | frequency | @ 3 GHz | @ 5 GHz | @ 9 GHz |
(n = 3.5) | (GHz) | (nH) | (nH) | (nH) |
HFSS simulation | 22.9 | 4.27 | 4.38 | 4.92 |
The model of the | 23.9 | 4.13 | 4.25 | 4.74 |
present invention | ||||
Greenhouse | X | 4.28 | 4.28 | 4.28 |
based model | ||||
Symbol definition: | ||||
(1) X represents NOT available. | ||||
(2) In the analysis, the geometry of the inductor is designed as lmax = 300 μm, s = 5 μm, and = 15 μm, respectively. The material utilized here is copper with the properties of ne = 8 × 1028 m−3, me = 9.11 × 10−31 kg, and σ0 = 5.6 × 107 (Ωm)−1 |
TABLE 3 |
COMPARISON OF INDUCTANCE |
OF RECTANGULAR INDUCTOR |
Self-resonant | Inductance | Inductance | Inductance | |
Comparisons | frequency | @ 3 GHz | @ 5 GHz | @ 9 GHz |
(n = 5.5) | (GHz) | (nH) | (nH) | (nH) |
HFSS simulation | 19.4 | 6.01 | 6.21 | 7.23 |
The model of the | 19.9 | 5.95 | 6.20 | 7.30 |
present invention | ||||
Greenhouse | X | 5.60 | 5.60 | 5.60 |
based model | ||||
The inductance expression based on the model of the present invention is closely fitted with the simulation and experimental data for the structure of the spiral inductor with substrate removal. The above table also indicated that not alike the model of the present invention the greenhouse model does not provide the self-resonant frequency itself and can not determine the inductances associated with the frequency change. The comparison of inductance spectrum is shown as
- 1. The present invention can accurately predict inductance and self-resonant frequency of spiral inductor base on physical and mathematical method.
- 2. The present invention is a pioneer in providing a method to evaluate the self-resonant frequency of a spiral inductor.
- 3. The present invention can evaluate the inductance and self-resonant frequency associated with altering material of the spiral inductors.
- 4. The present invention can describe the behaviors of electrons when suffering electromagnetic field in a metal with solid state physical sense.
- 5. The present invention can evaluate the energies stored in comers in the spiral inductors.
- 6. By means of Kramers-Kronig Relations, field theory, and energy conservation, the present invention can describe the exchange of the electromagnetic field when the self-resonant frequency occurs.
- 7. The present invention can describe the difference of the stored energy associated with different type of inductors; such as rectangular, octagonal, and circular inductors.
- 8. For different corner angle of the polygon spiral inductors, the present invention can evaluate the stored energy in the corners of rectangular, octagonal, circular inductors, respectively.
- 9. The present invention can evaluate require results rapidly with simple computer calculation. For instance, this model in comparison with the Ansoft-HFSS simulation can have less CPU processing time which is about 6400 times difference while both analyses are performed by the computer with the equipments of 3.4 GHz double CPUs and 2048 MB DDR2 Rams.
- 10. Scientists can evaluate the stored energies and behaviors of the electrons with their physical intuition.
Claims (18)
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TW095106923A TWI318365B (en) | 2006-03-02 | 2006-03-02 | A method for predicting inductance and self-resonant frequency of a spiral inductor |
TW95106923A | 2006-03-02 | ||
TW95106923 | 2006-03-02 |
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US20070214442A1 US20070214442A1 (en) | 2007-09-13 |
US7451415B2 true US7451415B2 (en) | 2008-11-11 |
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Citations (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US6311145B1 (en) * | 1999-06-17 | 2001-10-30 | The Board Of Trustees Of The Leland Stanford Junior University | Optimal design of an inductor and inductor circuit |
US6588002B1 (en) | 2001-08-28 | 2003-07-01 | Conexant Systems, Inc. | Method and system for predictive layout generation for inductors with reduced design cycle |
US6665849B2 (en) | 1999-06-09 | 2003-12-16 | Interuniversitair Microelektronica Centrum Vzw | Method and apparatus for simulating physical fields |
-
2006
- 2006-03-02 TW TW095106923A patent/TWI318365B/en not_active IP Right Cessation
- 2006-05-18 US US11/435,710 patent/US7451415B2/en not_active Expired - Fee Related
Patent Citations (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US6665849B2 (en) | 1999-06-09 | 2003-12-16 | Interuniversitair Microelektronica Centrum Vzw | Method and apparatus for simulating physical fields |
US6311145B1 (en) * | 1999-06-17 | 2001-10-30 | The Board Of Trustees Of The Leland Stanford Junior University | Optimal design of an inductor and inductor circuit |
US6588002B1 (en) | 2001-08-28 | 2003-07-01 | Conexant Systems, Inc. | Method and system for predictive layout generation for inductors with reduced design cycle |
Non-Patent Citations (6)
Title |
---|
Ansoft HFSS, 9.0 Version, Ansoft, http://www.ansoft.com/products/hf. |
Asgaran, The 14th International Conference on Microelectronics, Dec. 2002, pp. 247-250. |
C. C. Chen et al. A Closed-Form Integral Model of Spiral Inductor Using the Kramers-Kronig Relations, IEEE Microwave and Wireless Components Letters, vol. 15, No. 11 (Nov. 2005), pp. 778-780. * |
Greenhouse, IEEE Transaction on Parts, Hybrids, and Packaging, Jun. 1974, pp. 101-109, vol. PHP-10, No. 2. |
Jenei et al. IEEE Journal of Solid-State Circuits, Jan. 2002, pp. 77-80, vol. 37, No. 1. |
Mohan et al., IEEE Journal of Solid-State Circuits, Oct. 1999, pp. 1419-1424, vol. 34. |
Also Published As
Publication number | Publication date |
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TWI318365B (en) | 2009-12-11 |
US20070214442A1 (en) | 2007-09-13 |
TW200734927A (en) | 2007-09-16 |
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